# How do you simplify (-2/3)^-3?

Nov 7, 2015

$- \frac{27}{8}$

#### Explanation:

A negative exponent is solved in two steps: first of all, consider the inverse of your fraction (i.e. : change numerator and denominator), and then consider the positive power of this inverse fraction.

So, to compute your power, we make a first step:

${\left(- \frac{2}{3}\right)}^{-} 3 \to {\left(- \frac{3}{2}\right)}^{3}$

And then apply the usual rule for the power of a fraction, i.e. the power of the numerator divided by the power of the denominator:

${\left(- \frac{3}{2}\right)}^{3} = \left(- \frac{3}{2}\right) \cdot \left(- \frac{3}{2}\right) \cdot \left(- \frac{3}{2}\right) = - {3}^{3} / {2}^{3} = - \frac{27}{8}$

Nov 7, 2015

• Raise both the numerator and denominator to the power of (-3)
• Make the exponents positive

The exact value will be: $- \frac{27}{8}$

#### Explanation:

Before you start, it is a good idea to valuate the negative fraction. If you had raised the fraction to the power of an even number, the fraction would be positive. But since you are raising it to an odd number, the fraction will remain negative. Therefore, we will just keep the minus in the back of our heads.

Let's rewrite the fraction to make it a bit easier:
$- {\left(\frac{2}{3}\right)}^{-} 3$

We are allowed to do this since the exponent is an odd number.

We can use this rule to move the exponent into the fraction: ${\left(\frac{a}{b}\right)}^{x} = \left({a}^{x} / {b}^{x}\right)$

This gives us $- \left({2}^{-} \frac{3}{3} ^ - 3\right)$

Then we use a rule for negative exponents:
${a}^{-} x = \frac{1}{a} ^ x$

Because of the rule above, we can move ${2}^{-} 3$ down, and ${3}^{-} 3$ up;

$- \left({3}^{3} / {2}^{3}\right)$
$- \frac{27}{8}$

REMEMBER!!: We can ONLY put the minus sign outside of the fraction IF we are raising it to an odd number!